Calculus Examples

$\int_{0}^{3} 3 x - x^{2} d x$

Step 1

Split the single integral into multiple integrals.

$\int_{0}^{3} 3 x d x + \int_{0}^{3} - x^{2} d x$

Step 2

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int_{0}^{3} x d x + \int_{0}^{3} - x^{2} d x$

Step 3

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$3 (\frac{1}{2} x^{2}]_{0}^{3}) + \int_{0}^{3} - x^{2} d x$

Step 4

Combine $\frac{1}{2}$ and $x^{2}$ .

$3 (\frac{x^{2}}{2}]_{0}^{3}) + \int_{0}^{3} - x^{2} d x$

Step 5

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$3 (\frac{x^{2}}{2}]_{0}^{3}) - \int_{0}^{3} x^{2} d x$

Step 6

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$3 (\frac{x^{2}}{2}]_{0}^{3}) - (\frac{1}{3} x^{3}]_{0}^{3})$

Step 7

Simplify the answer.

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Step 7.1

Combine $\frac{1}{3}$ and $x^{3}$ .

$3 (\frac{x^{2}}{2}]_{0}^{3}) - (\frac{x^{3}}{3}]_{0}^{3})$

Step 7.2

Substitute and simplify.

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Step 7.2.1

Evaluate $\frac{x^{2}}{2}$ at $3$ and at $0$ .

$3 ((\frac{3^{2}}{2}) - \frac{0^{2}}{2}) - (\frac{x^{3}}{3}]_{0}^{3})$

Step 7.2.2

Evaluate $\frac{x^{3}}{3}$ at $3$ and at $0$ .

$3 (\frac{3^{2}}{2} - \frac{0^{2}}{2}) - (\frac{3^{3}}{3} - \frac{0^{3}}{3})$

Step 7.2.3

Simplify.

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Step 7.2.3.1

Raise $3$ to the power of $2$ .

$3 (\frac{9}{2} - \frac{0^{2}}{2}) - (\frac{3^{3}}{3} - \frac{0^{3}}{3})$

Step 7.2.3.2

Raising $0$ to any positive power yields $0$ .

$3 (\frac{9}{2} - \frac{0}{2}) - (\frac{3^{3}}{3} - \frac{0^{3}}{3})$

Step 7.2.3.3

Cancel the common factor of $0$ and $2$ .

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Step 7.2.3.3.1

Factor $2$ out of $0$ .

$3 (\frac{9}{2} - \frac{2 (0)}{2}) - (\frac{3^{3}}{3} - \frac{0^{3}}{3})$

Step 7.2.3.3.2

Cancel the common factors.

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Step 7.2.3.3.2.1

Factor $2$ out of $2$ .

$3 (\frac{9}{2} - \frac{2 \cdot 0}{2 \cdot 1}) - (\frac{3^{3}}{3} - \frac{0^{3}}{3})$

Step 7.2.3.3.2.2

Cancel the common factor.

$3 (\frac{9}{2} - \frac{2 \cdot 0}{2 \cdot 1}) - (\frac{3^{3}}{3} - \frac{0^{3}}{3})$

Step 7.2.3.3.2.3

Rewrite the expression.

$3 (\frac{9}{2} - \frac{0}{1}) - (\frac{3^{3}}{3} - \frac{0^{3}}{3})$

Step 7.2.3.3.2.4

Divide $0$ by $1$ .

$3 (\frac{9}{2} - 0) - (\frac{3^{3}}{3} - \frac{0^{3}}{3})$

Step 7.2.3.4

Multiply $- 1$ by $0$ .

$3 (\frac{9}{2} + 0) - (\frac{3^{3}}{3} - \frac{0^{3}}{3})$

Step 7.2.3.5

Add $\frac{9}{2}$ and $0$ .

$3 (\frac{9}{2}) - (\frac{3^{3}}{3} - \frac{0^{3}}{3})$

Step 7.2.3.6

Combine $3$ and $\frac{9}{2}$ .

$\frac{3 \cdot 9}{2} - (\frac{3^{3}}{3} - \frac{0^{3}}{3})$

Step 7.2.3.7

Multiply $3$ by $9$ .

$\frac{27}{2} - (\frac{3^{3}}{3} - \frac{0^{3}}{3})$

Step 7.2.3.8

Raise $3$ to the power of $3$ .

$\frac{27}{2} - (\frac{27}{3} - \frac{0^{3}}{3})$

Step 7.2.3.9

Cancel the common factor of $27$ and $3$ .

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Step 7.2.3.9.1

Factor $3$ out of $27$ .

$\frac{27}{2} - (\frac{3 \cdot 9}{3} - \frac{0^{3}}{3})$

Step 7.2.3.9.2

Cancel the common factors.

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Step 7.2.3.9.2.1

Factor $3$ out of $3$ .

$\frac{27}{2} - (\frac{3 \cdot 9}{3 (1)} - \frac{0^{3}}{3})$

Step 7.2.3.9.2.2

Cancel the common factor.

$\frac{27}{2} - (\frac{3 \cdot 9}{3 \cdot 1} - \frac{0^{3}}{3})$

Step 7.2.3.9.2.3

Rewrite the expression.

$\frac{27}{2} - (\frac{9}{1} - \frac{0^{3}}{3})$

Step 7.2.3.9.2.4

Divide $9$ by $1$ .

$\frac{27}{2} - (9 - \frac{0^{3}}{3})$

Step 7.2.3.10

Raising $0$ to any positive power yields $0$ .

$\frac{27}{2} - (9 - \frac{0}{3})$

Step 7.2.3.11

Cancel the common factor of $0$ and $3$ .

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Step 7.2.3.11.1

Factor $3$ out of $0$ .

$\frac{27}{2} - (9 - \frac{3 (0)}{3})$

Step 7.2.3.11.2

Cancel the common factors.

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Step 7.2.3.11.2.1

Factor $3$ out of $3$ .

$\frac{27}{2} - (9 - \frac{3 \cdot 0}{3 \cdot 1})$

Step 7.2.3.11.2.2

Cancel the common factor.

$\frac{27}{2} - (9 - \frac{3 \cdot 0}{3 \cdot 1})$

Step 7.2.3.11.2.3

Rewrite the expression.

$\frac{27}{2} - (9 - \frac{0}{1})$

Step 7.2.3.11.2.4

Divide $0$ by $1$ .

$\frac{27}{2} - (9 - 0)$

Step 7.2.3.12

Multiply $- 1$ by $0$ .

$\frac{27}{2} - (9 + 0)$

Step 7.2.3.13

Add $9$ and $0$ .

$\frac{27}{2} - 1 \cdot 9$

Step 7.2.3.14

Multiply $- 1$ by $9$ .

$\frac{27}{2} - 9$

Step 7.2.3.15

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{27}{2} - 9 \cdot \frac{2}{2}$

Step 7.2.3.16

Combine $- 9$ and $\frac{2}{2}$ .

$\frac{27}{2} + \frac{- 9 \cdot 2}{2}$

Step 7.2.3.17

Combine the numerators over the common denominator.

$\frac{27 - 9 \cdot 2}{2}$

Step 7.2.3.18

Simplify the numerator.

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Step 7.2.3.18.1

Multiply $- 9$ by $2$ .

$\frac{27 - 18}{2}$

Step 7.2.3.18.2

Subtract $18$ from $27$ .

$\frac{9}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{9}{2}$

Decimal Form:

$4.5$

Mixed Number Form:

$4 \frac{1}{2}$

Step 9